Chapter 10: Conic Sections

A conic section is the curve you get by slicing a double cone with a plane. Depending on the angle of the slice, you get four families: circles, ellipses, parabolas, and hyperbolas. These four curves have been studied for over two thousand years , they describe the orbits of planets, the paths of projectiles, the shape of satellite dishes, and countless other physical situations.

Each conic has an elegant focus-directrix characterisation: a conic is the set of points $P$ whose distance from a fixed point (the focus) bears a constant ratio (the eccentricity $e$ ) to its distance from a fixed line (the directrix). When $e = 0$ it is a circle, $e < 1$ an ellipse, $e = 1$ a parabola, $e > 1$ a hyperbola.

In this chapter we focus on the standard placements: conics centred at the origin with axes along the coordinate axes. This keeps the algebra simple while displaying all the geometric features clearly. Each conic gets a few standard equations and a vocabulary , vertex, focus, directrix, axis, latus rectum, eccentricity.

For Class XII and JEE, conics return in the chapter on differential equations (planetary motion) and many optimisation problems. The standard forms you memorise now will serve for years.

This chapter is essentially geometry made algebraic. Every problem reduces to identifying which conic, in which form, and then plugging in the appropriate formula. Sketch the curve before computing , a picture saves five minutes of algebra.

What's inside

Sections of a cone and overview , how the four conics arise.
Circle , equation, centre, radius, general form.
Parabola , standard forms, focus, directrix, latus rectum.
Ellipse , standard form, foci, eccentricity, latus rectum.
Hyperbola , standard form, foci, asymptotes, eccentricity.
Comparison and applications , solving mixed problems.

Key results / Formula card

Conic	Standard form	Key parameters
Circle (centre origin)	$x^2 + y^2 = r^2$	radius $r$
Circle (centre $(h, k)$ )	$(x - h)^2 + (y - k)^2 = r^2$	centre, radius
Parabola (right-opening)	$y^2 = 4 a x$	focus $(a, 0)$ , directrix $x = -a$ , latus rectum $4a$
Parabola (up-opening)	$x^2 = 4 a y$	focus $(0, a)$ , directrix $y = -a$
Ellipse	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, a > b$	$c^2 = a^2 - b^2$ , foci $(\pm c, 0)$ , $e = c/a < 1$
Hyperbola	$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$	$c^2 = a^2 + b^2$ , foci $(\pm c, 0)$ , $e = c/a > 1$
Asymptotes of hyperbola	$y = \pm (b/a) x$
Latus rectum of ellipse/hyperbola	$\dfrac{2 b^2}{a}$

How to read this chapter

For every problem: (1) identify the conic and orientation, (2) write the standard form, (3) read off the parameters, (4) check by sketching. The four conics are siblings , once you know one, the others fall in line.

Chapter 10: Conic Sections

What's inside

Key results / Formula card

How to read this chapter

Sub-topics

Practice quiz